Mathematics of Planet Earth

The following sections are included:

• Foreword
• Introduction
• Contents

The following sections are included:

• Introduction
• Zonally Integrated View of Atmospheric and Oceanic Circulation
  • Energy balance
  • Energy equation
  • Fluxes
  • Hadley circulation
  • Moisture budget
  • Angular momentum budget
• The Wind-Driven Ocean Circulation
  • Ekman transport
  • Regions of systematic oceanic upwelling
  • Western boundary currents
• Stationary Rossby Waves in the Atmosphere
  • Theory
  • Some principal stationary wave sources
• Transient Disturbances in the Atmosphere
  • Extratropical storms
  • Predictability
  • Tropical dynamics
  • Eddy momentum fluxes
• Climate Change
  • Global-mean warming
  • Regional aspects of climate change
• Appendix
  • Hydrostatic balance
  • Thermal-wind balance
  • Adiabatic relations

The following sections are included:

• Preliminaries
  • Grad, Div, and Curl
  • The Levi-Civita symbol
  • Differential form notation for vector calculus
  • What is a Partial Differential Equation (PDE)?
  • General properties of solutions to PDEs — Symmetry and similarity
• The Governing Equations for an Ideal Fluid
  • The governing equations
  • Rotation and vorticity formulation of the Euler equation
• Linear and Nonlinear Transport in One-Dimensional Flows
• Nonlinear Diffusion
  • General solutions: Linearising transformation
• Reductions of the Euler Equations for Modelling Large-Scale Flows
  • Geostrophic reductions
• Rotating Shallow Water (RSW) Equations
  • Rotating shallow water motion equations
  • Geostrophic balance
  • Crucial vector calculus identities
  • Conservation laws for RSW
• The Quasigeostrophic (QG) Approximation
  • Derivation of QG
  • The ageostrophic velocity in the QG approximation: Part I
  • The ageostrophic velocity in the QG approximation: Part II
  • Rossby waves for QG and their dispersion relation
• Fundamental Conservation Laws of the QG Equations
  • Energy and circulation
  • Potential vorticity
  • Casting QG into Hamiltonian form
• Equilibrium Solutions of QG
  • Critical point solutions of QG
• Alternative Representations of RSW: Part I
  • Vorticity, divergence and depth representation
• Alternative Representations of RSW: Part II
  • Slow + fast decomposition
• Hamilton’s Principle for Simple Ideal Fluids
  • Preparation for fluid dynamical variational principles
  • Explicitly varying the action integral
  • Deriving the Euler–Poincaré motion equation
• Hamilton’s Principle for RSW
  • Properties of the Euler–Poincaré motion equation
  • Specialising to the RSW equation
  • Hamiltonian formulation of the Euler–Poincaré equations
  • Thermal rotating shallow water

The following sections are included:

• Introduction
  • Probability
  • Why the classical interpretation of probability is insufficient?
  • Statistics
• Sigma Algebras and Probability Measures
• Measurable Functions and Integration
  • The integral
  • Interchange of integral with a.s. limits
• Transformations
• Products Spaces and Product Measures, Fubini-Theorem
• Distributions and Independence
  • Independence
  • Modelling with random variables
• Introduction to Statistics
• Conditional Probabilities and Conditional Expectations
  • Regular conditional probabilities
• Convergence in Distribution and Characteristic Functions
  • The Central Limit Theorem
• Introduction to Bayesian Computation
  • Setting up the statistical framework
  • Should we employ frequentist or Bayesian approaches?
• Bayesian Inference
  • Conjugate priors
  • Posterior summaries
• Monte Carlo Integration
  • Variance reduction techniques
• Simulating Random Variables
  • Transformation methods
  • Rejection sampling
  • Importance sampling
• Markov Chain Monte Carlo
• Literature on Measure Theory and Integration
• Solutions to Selected Exercises

The following sections are included:

• Introduction
• Homeomorphisms of the Circle
  • Rigid rotations
  • Distribution of orbits
  • Homeomorphisms of the circle
• Expanding Maps of the Circle
• Symbolic Dynamics
  • Coding expanding maps of the circle
  • Coding horseshoe maps
• Topological Entropy
• Ergodic Theory
  • Recurrence
  • Kac Lemma
  • Birkhoff’s Ergodic Theorem
  • Ergodicity and mixing
• Transfer Operator Techniques
  • Metrics on cones
  • Transfer operator
  • Existence of invariant measure
  • Decay of correlation
  • Central Limit Theorem
  • Linear response

The following sections are included:

• Introduction and the Navier–Stokes Equations
• Modelling Advection
  • Linear finite difference schemes
  • Order of accuracy and order of convergence
  • Implicit and explicit numerical methods
  • Some definitions of numerical properties
  • The Lax Equivalence Theorem
  • The domain of dependence and the Courant–Friedrichs–Lewy (CFL) criterion
  • Von Neumann stability analysis
  • Dispersion errors
  • Conservation
• The Shallow Water Equations
  • Finite difference schemes for the one-dimensional, linearised shallow water equations
• Iterative Methods for Linear Systems in the Geosciences
• Time-Stepping Algorithms for the One-Dimensional Shallow Water Equations
• The Conjugate Gradient Algorithm
• Convergence of CG
• Preconditioning
• Classical Iterative Methods as Preconditioners
  • Richardson’s method
  • Jacobi’s method
  • Gauss–Seidel method
• Using Splitting Methods as Preconditioners
• Convergence Properties for Splitting Methods
• Convergence Analysis for Splitting Methods
  • Richardson iteration
  • Convergence analysis for symmetric matrices

The following sections are included:

• Bibliography
• Index